A new estimator of the self-similarity exponent through the empirical likelihood ratio test

A new method is proposed to estimate the self-similarity exponent. Instead of applying finite moment(s) methods, a goodness-of-fit statistic is designed to test whether two rescaled sequences are drawn from the same distribution, which is the definition of self-similarity. The test is the empirical likelihood ratio, which is robust with respect to processes with dependence. We provide a closed formula for fractional Brownian motion and prove that the distance between two rescaled sequences is zero iff the scaling exponent equals the true one. According to our results, the method we propose can identify self-similarity and effectively estimate the corresponding exponent.